Design Optimization Considering Performance And Reliability

Design Optimization Considering Performance And Reliability
Shengkui Zeng, PhD, Beihang University
Jiming Ma, PhD, Beihang University
Feixia Li, M.S, Beihang University
Key Words: Electro-Hydrostatic Actuator (EHA), integrated design, optimization, performance, reliability
SUMMARY & CONCLUSIONS
Current design and optimization of a flight control
actuator system is based on a traditional design theory and
sophisticated methods to get stable and good performance
ability, but designed scheme may be sensitive to the
environment fluctuation, internal parameter variation and
component catastrophic failure.
Reliability of actuator system is the ability that system
can satisfactorily implement its defined function. There are
two kinds of factors influencing actuator reliability, one is
component catastrophic failure; the other is disturbance from
internal parameters and environment variations. Both factors
may cause performance degradation (soft failure) and even
result in mission abort (hard failure). In a redundant actuator
system, component catastrophic failure will result in more
serious performance degradation, even if it cannot cause hard
failure. In this research, in order to improve the reliability and
performance simultaneously, we have developed an integrated
design optimization method which integrates performance and
reliability requirements into the design process.
A
mathematical model for an Electro-Hydrostatic Actuator
(EHA) has been built first and then injected with uncertain
characters of parameters variations, component failure and
disturbance from environment based on their distributions and
mission profile. After that, a response surface between
reliability and system Design Dependent Parameters (DDPs)
has been created based on the simulation performance results,
thus producing optimized DDPs. The simulation results have
also shown that this method is feasible.
1 INTRODUCTION
A design completed using traditional methods may NOT
always be robust and reliable [1]. Based on binary component
status, conventional reliability design and analysis methods
focus primarily on statistically evaluation of product failure
times and weaknesses.
Performance and reliability
requirements are considered separately in design process
[2,3,4], and cannot get an integrated optimization result.
Current fault tolerant control system design method has
considered the effect of noise to the performance [5,6], but the
quantitative reliability requirements are rarely constrains in the
design process, therefore, it is difficult to meet both reliability
and performance requirements. It has been shown in the
1-4244-2509-9/09/$20.00 ©2009 IEEE
literature that integrated analysis considering performance and
reliability can create relationship between soft and hard failure
[7,8,9,10,11]. Based on these results, we have developed an
integrated design optimization method.
The paper is
organized as follows: Section 3 introduces a mathematical
model of EHA and its performance and reliability
requirements. Section 4 presents the approach and destination
of integrated design optimization, and how simulation results
are used to create the response surface between reliability and
system DDPs. Section 5 presents details of the EHA case
study where reliability response surface is created, and
optimized DDPs and system reliability are achieved.
2 NOTATIONS & ACRONYMS
Ce (V/(rad.s-1)): Back electromotive force constant
Kp: Proportional gain of PID controller
Tr (s): Rising time
Ts (s): Settling time
Ess (m): Steady state error
Pos: Percent overshoot
F_F (N) :Fighting force
J (kg.m2): Moment of inertia of motor&pump
R (Ω): Armature resistance
L (H): Armature inductance
Cvf (Nm/(rev.min-1)): Coefficient of viscous friction of motor
ηv: Volumetric efficiency of pump
Bd (N/(m.s-1)): Damper rating of load
KL (N/m): Spring rate of load
Dp (ml/rev): Pump displacement
Ve (m/s): Aircraft velocity in the earth-fixed reference frame
ρ (kg/m3): Air density
λAmp: Failure rate of amplifier
λs: Failure rate of sensor
λPower: Failure rate of power unit
λAct: Failure rate of actuator
λPump: Failure rate of pump
FCC: Flight Control Computer
LVDT: Linear Variable Displacement Transducer
3 DESIGN OPTIMIZATION REQUIREMENT OF EHA
3.1 Quadruple EHA Structure
EHA is defined as an electro-hydrostatic actuator which
incorporates an electric motor driving a hydraulic pump, a
Control
Electronics 1
(PID)
Xcmd
(From FCC)
Control
Electronics 2
(PID)
Armature 1
Amplifier 1
Cylinder 1
Motor 1
Pump 1
M
Armature 2
Amplifier 2
Control
Surface
Armature 3
Control
Electronics 3
(PID)
Amplifier 3
Control
Electronics 4
(PID)
Amplifier 4
Cylinder 2
Motor 2
Pump 2
M
Quadruple
LVDT
Voting Armature 4
Mode
Figure 1 - Simplified Quadruple Redundancy EHA
hydraulic fluid reservoir, a servo-actuator and other necessary
accessories packaged in a single, self-contained Line Replace
Unit (LRU). Figure 1 shows a quadruple EHA which has
Failure-Operable (FO) / FO / Failure-Safety (FS) ability
according to flight control system requirements [12].
3.2 Performance and Reliability Requirements
Performance requirements of EHA focus on stability,
response time, accuracy and fighting forces between different
channels. Reliability requirement is allocated from a Flight
Control System (FCS). The EHA has two kind of failure
(hard and soft failure), and the effects of both kind of failures
can be described through performance characteristics. Figure
2 shows step response of EHA when a cylinder leakage
resulted in loss of redundancy.
Figure 2 - Loss of Redundancy Results in
Performance Degradation
Performance is influenced by many factors, such as
Design Depend Parameters (DDPs), internal parameters
variation, environment fluctuations, and component failures.
This paper presents a design framework that can discover the
quantitative relationship between reliability and DDPs based
on the performance criteria.
The model parameters
uncertainties are represented by statistical distributions. As
the most frequently used signal source, step response is
selected as the criteria for performance characters. Table 1
shows performance requirements for a step response.
Performance requirements
(Step response)
Reliability
Tr*
Ts*
Ess*
Pos*
F_F*
<0.3
<1
<1.5e-3
<15%
<300
As high as
possible
Table 1 – Performance & Reliability Requirements
of EHA (Full Stroke)
4 AN INTEGRATED DESIGN OPTIMIZATION
The goal of an integrated design optimization is high
reliability and optimized DDPs, which meet the system
performance requirements.
Reliability response surface method is the key element of
an integrated design optimization. As a function of DDPs, it is
created by integrated simulation and data regression. During
the integrated simulation program, the environment
fluctuations, internal parameters variations and component
catastrophic failures are provided into the EHA performance
model to evaluate their impact. Taking reliability as a target
and performance requirements as constrains, the optimization
process has been able to evaluate the performance & reliability
quantitative requirements simultaneously.
This research on EHA uses back electromotive force
constant (Ce) and proportional gain of PID controller (Kp) as
the DDPs, with the goal of getting optimized DDPs
considering all the uncertain factors.
Design process includes the following two steps:
4.1 Creating reliability response surface
Figure 3 shows the scheme of creating the response
surface. A model in this scheme has been exposed to all kinds
of uncertain parameters effects (noise).
In this model, we assume perfect Fault Diagnosis and
Isolation (FDI) ability, and U are the input parameters of the
system, Y are the performance parameters achieved through
simulation, S are performance parameters requirement domain,
Xi+1-Xn
Noise
U
X1-Xi
Controller
Noise
Noise
Noise
Load
Plant
Z1-Zk
Noise
Noise
Y
S
P (Y ⊂ S )
R
R = g (U )
Sensor
5 SIMULATION, CREATING RESPONSE SURFACE
AND RESULTS
Simulation analysis is based on the mathematical model
created in Matlab and AMESim. During the simulation, the
component catastrophic failures and the environment
condition varies within the mission profile are introduced into
the model. The EHA uncertainties distributions and are
shown in Table 2.
Figure 3 - Response Surface Creating Scheme
and R is the system reliability. Internal uncertain
parameters (X1-Xi), environment varying factors (Xi+1-Xn) and
component failures (Z1-Zk) have been considered during the
simulation process.
Y = f (U , X , Z )
(1)
We assume that X1-Xi and Y follow normal probability
distribution, and Z follow exponential distribution. During the
simulation process, these X and Z are the statistical constants.
If the simulation is repeated for a sufficient number of times,
Y will be convergent based on its natural distribution. Then
we can get:
( μY ，σ Y 2 ) = f (U ) | X ~ N ( μ X ,σ 2 X ), Z ~ e(λZ ) (2)
Where μY , σ Y are mean and variance of Y. Given μY , σ Y ,
then we get the R.
R = P (Y > s )
(3)
Where
s: the performance criteria.
In this case, Y have normal distribution, then we have:
(4)
R = Φ[(μY − s) / σ Y ]
After sufficient times of simulation, we can get the response
surface as follows:
R = g (U )
(5)
4.2 Integrated Design Optimization Model Considering
Performance and Reliability
In the optimization model, maximum reliability is the
optimization target, performance Y* are the constrains, and
DDPs (Ce, Kp) are the design variables. Final goal is getting
optimized DDPs, which can realize highest reliability and
meet the performance requirements. Optimization follows the
regulation as equations (6).
Max
R
S.t
Tr-Tr*<0
Ts-Ts*<0
Pos-Pos*<0
D.V. Ce,Kp
(6)
Tr,Ts,Pos are the performance parameters without component
failure and uncertainty. The relation between DDPs and these
parameters can be achieved through simulation and analysis.
Tr*,Ts*,Pos* are performance requirements for EHA step
response (see Table 1). Ess and F_F are very near to zero
without disturbance, therefore, these two parameters are not
set as constrains in equations (6).
X1-8
X9-10
Z
Symbol
X1
X2
X3
X4
Notation
J
R
L
Cvf
~N(1.6E-3,2E-4)
~N(0.5,0.06)
~N(1E-2,9E-4)
~N(3E-4,2E-5)
X5
ηv
~N(0.85,0.05)
X6
X7
X8
X9
X10
Z1
Z2
Z3
Z4
Z5
Bd
KL
Dp
ρ
Ve
λAmp
λS
λPower
λAct
λPump
~N(1000,50)
~N(5E6,1E5)
~N(1,0.08)
Value
Distribution
Normal
Decided by Mission Profile (MP)
~e( 45E-6)
~e( 15E-6)
~e (26E-6)
~e( 4E-6)
~e (12E-6)
Exponential
Table 2 - Uncertain Parameters in EHA Model
The design optimization process is shown in Figure 4.
This analysis used 100 DDPs values, with 1000 simulations
for each DDPs under the varying internal and environmental
conditions. After 1000 simulations, we obtain performance
values (Y) with the mean and variance of Y( μY , σ Y ). Then,
we get a reliability value (R) corresponding to the current
DDPs based on equation (4). After 100×1000 simulation
times, we get the response surface [13] based on simulation
results:
R = a0 + a1Ce − a2 Kp − a3Ce2 + a4 Kp2 + CeKp(a5 + a6Ce − a7 Kp) (7)
Where: a0=1.018999, a1=6.16564E-3, a2=2.38192E-4,
a3=0.5402, a4=6.03544E-7, a5=8.47214E-4, a6=1.94068E-3,
a7=3.09908E-6.
According to optimization regulation of equations (6),
based on Zoutendijk algorithm [14], we obtain the integrated
design optimization results, DDPs, reliability and performance
characters as shown in Table 3.
6 CONCLUDING REMARKS
Both performance and reliability of the EHA are closely
related with DDPs. To obtain an optimized design, we must
consider both the performance and reliability requirements
during the design process.
response surface between EHA’s reliability and DDPs
(Ce,Kp). Then, we created an optimization model to get the
DDPs, which can deliver maximum reliability and meet the
performance requirements. Final results (DDPs) show higher
reliability and better performance than the original design.
Create performance
model
Select DDPs
(Ce,Kp)
REFERENCES
Experiment design
of DDPs
1.
Get feasible
DDP(DDP[1]-DDP[n])
2.
Select DDP[i]
3.
Set simulation
times: N
Inject
Environment
Noise(X9-X10)
Inject Internal
Noise (X1-X8)
Fault Injection
Based Failure
Rate(Z1-Z5)
Integrated
Simulation
No
4.
5.
6.
Simulation Times≥ N?
Yes
Get Performance Y[i]
(Tr,Ts,Ess,Pos,F_F)
Get
μ Y ,σ Y
7.
8.
Get R[i]
i=i+1
No
i≥ n?
Yes
Create Response Surface
R=g(DDPs)
9.
10.
Optimization regulation
Max R
S.t Y ⊂Y*
D.V. DDPs
Get Optimizated
DDPs & R
Figure 4 - Integrated Design Optimization Flow Diagram
Performance
Reliability
characters
Ce Kp Tr (s ) Ts (s) Pos
R
Original 0.215 200 0.34 0.39 20.98 0.99957
Optimized 0.235 264 0.30 0.34 15.18 0.99988
DDP
Table 3 - Performance and Reliability Results
Before and After Optimization
This paper described an EHA analysis model. After
simulation utilizing internal parameter variations, environment
fluctuations, and component catastrophic failures, we get a
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BIOGRAPHIES
Shengkui Zeng, Professor
Department of System Engineering
Beihang University
37# Xueyuan Street
Haidian District, Beijing, 100191, P.R.China
e-mail: [email protected]
Shengkui Zeng is a professor and vice-dean of Department of
System Engineering of Beihang University (BUAA). He has
been a visiting researcher in CALCE EPSC at University of
Maryland in 2005. His research and teaching interests are
reliability modeling and optimization, system reliability
simulation, Reliability Based Multidisciplinary Design
Optimization (RBMDO). He is a team lead for the KWARMS© reliability software platform, a co-author of three
books, and holder of three ministry level professional awards.
Jiming Ma, PhD
Department of System Engineering
Beihang University
37# Xueyuan Street
Haidian District, Beijing, 100191, P.R.China
e-mail: [email protected]
Jiming Ma serves as an associate professor in the Department
of System Engineering at BUAA. He earned his doctoral
degree in the School of Automation Science and Electrical
Engineering of BUAA in 2006. He earned his bachelor and
master degrees at BUAA during the period from 1996 to 2002.
At present, his research activities are centered on the
reliability improvement of electro-mechanical system,
integrated design considering performance and reliability of
complex equipment.
Feixia Li, M.S candidate
Department of System Engineering
Beihang University
37# Xueyuan Street
Haidian District, Beijing, 100191, P.R.China
e-mail: [email protected]
Feixia Li is studying in the Department of System Engineering
at BUAA as a master candidate. She earned bachelor degree
at Qingdao University in 2006. She conducts research on
reliability analytical and simulation-based methods.

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